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| In [[quantum physics]], a '''parity transformation''' (also called '''parity inversion''') is the flip in the sign of ''one'' [[Three-dimensional space|spatial]] [[coordinate]]. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:
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| :<math>\mathbf{P}: \begin{pmatrix}x\\y\\z\end{pmatrix} \mapsto \begin{pmatrix}-x\\-y\\-z\end{pmatrix}.</math>
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| It can also be thought of as a test for [[Chirality (physics)|chirality]] of a physical phenomenon, in that performing a parity inversion transforms a chiral phenomenon into its mirror image. A parity transformation on something achiral, on the other hand, can be viewed as an identity transformation. All fundamental interactions of [[elementary particle]]s are symmetric under parity, except for the [[weak interaction]], which is sensitive to chirality and thus provides a handle for probing it, elusive as it is in the midst of stronger interactions. In interactions which are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions.
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| A 3×3 matrix representation of '''P''' would have [[determinant]] equal to −1, and hence cannot reduce to a [[rotation]] which has a determinant equal to 1. The corresponding mathematical notion is that of a [[point reflection]].
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| In a two-dimensional plane, parity is ''not'' a simultaneous flip of all coordinates, which would be the same as a [[rotation]] by 180 degrees. It is important that the determinant of the P matrix be −1, which does not happen for 180 degree rotation in 2-D, where a parity transformation flips the sign of ''either x or y, but not both''.
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| ==Simple symmetry relations==
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| Under [[rotation]]s, classical geometrical objects can be classified into [[scalar (physics)|scalars]], [[Euclidean vector|vector]]s, and [[tensor]]s of higher rank. In [[classical physics]], physical configurations need to transform under [[group representation|representation]]s of every symmetry group.
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| [[quantum mechanics|Quantum theory]] predicts that states in a [[Hilbert space]] do not need to transform under representations of the [[group (mathematics)|group]] of rotations, but only under [[projective representation]]s. The word ''projective'' refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not an observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states.
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| The projective representations of any group are isomorphic to the ordinary representations of a [[Group extension#Central extension|central extension]] of the group. For example, [[projective representation]]s of the 3-dimensional rotation group, which is the [[special orthogonal group]] SO(3), are ordinary representations of the [[special unitary group]] SU(2) (see [[representation theory of SU(2)]]). Projective representations of the rotation group that are not representations are called [[spinor]]s, and so quantum states may transform not only as tensors but also as spinors.
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| If one adds to this a classification by parity, these can be extended, for example, into notions of
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| *''scalars'' (''P'' = 1) and ''[[pseudoscalar (physics)|pseudoscalar]]s'' (''P'' = −1) which are rotationally invariant.
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| *''vectors'' (''P'' = −1) and ''axial vectors'' (also called ''[[pseudovector]]s'') (''P'' = 1) which both transform as vectors under rotation.
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| One can define '''reflections''' such as
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| :<math>V_x: \begin{pmatrix}x\\y\\z\end{pmatrix} \mapsto \begin{pmatrix}-x\\y\\z\end{pmatrix},</math> | |
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| which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing ''x''-, ''y''-, and ''z''-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In odd number of dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used.
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| Parity forms the [[Abelian group]] ℤ<sub>2</sub> due to the relation '''P'''<sup>2</sup> = 1. All Abelian groups have only one dimensional [[irreducible representations]]. For ℤ<sub>2</sub>, there are two irreducible representations: one is even under parity ({{math|'''P'''''φ'' {{=}} ''φ''}}), the other is odd ({{math|'''P'''''φ'' {{=}} −''φ''}}). These are useful in [[quantum mechanics]]. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any [[phase (waves)|phase]].
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| ==Classical mechanics==
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| Newton's equation of motion {{math|'''F''' {{=}} ''m'' '''a''' }} (if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity.
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| However, angular momentum '''L''' is an [[axial vector]],
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| :'''L''' = '''r''' × '''p''',
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| :'''P'''('''L''') = (−'''r''') × (−'''p''') = '''L'''.
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| In classical [[electrodynamics]], the charge density ''ρ'' is a scalar, the electric field, '''E''', and current '''j''' are vectors, but the magnetic field, '''H''' is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector.
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| ==Effect of spatial inversion on some variables of classical physics==
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| ===Even===
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| Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:
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| :<math>\ t</math>, the [[time]] when an event occurs
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| :<math>\ m</math>, the [[mass]] of a particle
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| :<math>\ E</math>, the [[energy]] of the particle
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| :<math>\ P</math>, [[Power (physics)|power]] (rate of [[work (physics)|work]] done)
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| :<math>\ \rho</math>, the electric [[charge density]]
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| :<math>\ V</math>, the [[electric potential]] ([[volt]]age)
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| :<math>\ \rho</math>, [[energy density]] of the [[electromagnetic field]]
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| :<math>\mathbf L</math>, the [[angular momentum]] of a particle (both [[orbital motion|orbital]] and [[Spin (physics)|spin]]) (axial vector)
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| :<math>\mathbf B</math>, the [[magnetic field]] (axial vector)
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| :<math>\mathbf H</math>, the [[magnetic field#The Difference between B and H|auxiliary magnetic field]]
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| :<math>\mathbf M</math>, the [[magnetization]]
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| :<math>\ T_{ij}</math> [[Maxwell stress tensor]].
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| :All masses, charges, coupling constants, and other physical constants, except those associated with the weak force
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| ===Odd===
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| Classical variables, predominantly vector quantities, which have their sign flipped by spatial inversion include:
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| :<math>\ h</math>, the [[helicity (particle physics)|helicity]]
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| :<math>\ \Phi</math>, the [[magnetic flux]]
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| :<math>\mathbf x</math>, the [[position (vector)|position]] of a particle in three-space
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| :<math>\mathbf v</math>, the [[velocity]] of a particle
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| :<math>\mathbf a</math>, the [[acceleration]] of the particle
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| :<math>\mathbf p</math>, the [[linear momentum]] of a particle
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| :<math>\mathbf F</math>, the [[force (physics)|force]] exerted on a particle
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| :<math>\mathbf J</math>, the electric [[current density]]
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| :<math>\mathbf E</math>, the [[electric field]]
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| :<math>\mathbf D</math>, the [[electric displacement field]]
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| :<math>\mathbf P</math>, the [[electric polarization]]
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| :<math>\mathbf A</math>, the electromagnetic [[vector potential]]
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| :<math>\mathbf S</math>, [[Poynting vector]].
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| ==Quantum mechanics==
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| ===Possible eigenvalues===
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| [[Image:parity 1drep.png|thumb|200px|right|Two dimensional representations of parity are given by a pair of quantum states which go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all [[irreducible representation]]s of parity are one-dimensional.]]
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| In [[quantum mechanics]], spacetime transformations act on [[quantum states]]. The parity transformation, '''P''', is a [[unitary operator]], in general acting on a state ψ as follows: '''P'''ψ(''r'') = ''e''<sup>''iφ''/2</sup>ψ(−''r'').
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| One must then have '''P'''<sup>2</sup>ψ(''r'') = ''e''<sup>''iφ''</sup>ψ(''r''), since an overall phase is unobservable. The operator '''P'''<sup>2</sup>, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases ''e''<sup>''iφ''</sup>. If '''P'''<sup>2</sup> is an element ''e''<sup>''iQ''</sup> of a continuous U(1) symmetry group of phase rotations, then ''e''<sup>−''iQ''/2</sup> is part of this U(1) and so is also a symmetry. In particular, we can define '''P′''' = '''P'''''e''<sup>−''iQ''/2</sup>, which is also a symmetry, and so we can choose to call '''P′''' our parity operator, instead of '''P'''. Note that '''P′'''<sup>2</sup> = 1 and so '''P′''' has eigenvalues ±1. However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ±1.
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| ===Consequences of parity symmetry===
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| When parity generates the [[Abelian group]] ℤ<sub>2</sub>, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number
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| In quantum mechanics, [[Hamiltonian (quantum mechanics)|Hamiltonian]]s are [[Invariant (physics)|invariant]] (symmetric) under a parity transformation if '''P''' [[commutator|commutes]] with the Hamiltonian. In non-relativistic [[quantum mechanics]], this happens for any potential which is scalar, i.e., ''V'' = ''V''(''r''), hence the potential is spherically symmetric. The following facts can be easily proven:
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| *If |''A''{{rangle}} and |''B''{{rangle}} have the same parity, then {{langle}}''A''| '''X''' |''B''{{rangle}} = 0 where '''X''' is the [[position operator]].
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| *For a state |'''L''', ''L''<sub>z</sub>{{rangle}} of orbital angular momentum '''L''' with z-axis projection ''L''<sub>z</sub>, '''P'''|'''L''', ''L''<sub>z</sub>{{rangle}} = (−1)<sup>''L''</sup>|'''L''', ''L''<sub>z</sub>{{rangle}}.
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| *If ['''H''', '''P'''] = 0, then atomic dipole transitions only occur between states of opposite parity.<ref>
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| {{cite book
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| |last1=Bransden |first1=B. H. | |
| |last2=Joachain |first2=C. J.
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| |year=2003
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| |title=Physics of Atoms and Molecules
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| |edition=2nd |page=204
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| |publisher=[[Prentice Hall]]
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| |isbn=978-0-582-35692-4
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| }}</ref>
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| *If ['''H''', '''P'''] = 0, then a non-degenerate eigenstate of '''H''' is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of '''H''' is either invariant to '''P''' or is changed in sign by '''P'''.
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| Some of the non-degenerate eigenfunctions of '''H''' are unaffected (invariant) by parity '''P''' and the others will be merely reversed in sign when the Hamiltonian operator and the parity operator [[commutator|commute]]:
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| ::'''P''' Ψ= ''c'' Ψ,
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| where ''c'' is a constant, the [[eigenvalue]] of '''P''',
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| ::'''P'''<sup>2</sup>Ψ = ''c'''''P''' Ψ.
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| ==Quantum field theory==
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| :''The intrinsic parity assignments in this section are true for relativistic quantum mechanics as well as quantum field theory.''
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| If we can show that the [[vacuum state]] is invariant under parity ('''P'''|0{{rangle}} = |0{{rangle}}), the Hamiltonian is parity invariant (['''H''', '''P'''] = 0) and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction.
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| To show that [[quantum electrodynamics]] is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume that [[canonical quantization]] is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell's equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator:
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| :'''Pa'''('''p''', ±)'''P'''<sup>+</sup> = −'''a'''(−'''p''', ±)
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| where '''p''' denotes the momentum of a photon and ± refers to its polarization state. This is equivalent to the statement that the photon has odd [[intrinsic parity]]. Similarly all [[vector boson]]s can be shown to have odd intrinsic parity, and all [[pseudovector meson|axial-vectors]] to have even intrinsic parity.
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| There is a straightforward extension of these arguments to scalar field theories which shows that scalars have even parity, since
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| :'''Pa'''('''p''')'''P'''<sup>+</sup> = '''a'''(−'''p''').
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| This is true even for a complex scalar field. (''Details of [[spinor]]s are dealt with in the article on the ''[[Dirac equation]]'', where it is shown that [[fermion]]s and antifermions have opposite intrinsic parity.'')
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| With [[fermion]]s, there is a slight complication because there is more than one [[spin group]].
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| ==Parity in the standard model==
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| ===Fixing the global symmetries===
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| {{See also|(−1)F|l1=(−1)<sup>''F''</sup>}}
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| In the [[Standard Model]] of fundamental interactions there are precisely three global internal [[U(1)]] symmetry groups available, with charges equal to the [[baryon]] number ''B'', the [[lepton]] number ''L'' and the [[electric charge]] ''Q''. The product of the parity operator with any combination of these rotations is another parity operator. It is conventional to choose one specific combination of these rotations to define a standard parity operator, and other parity operators are related to the standard one by internal rotations. One way to fix a standard parity operator is to assign the parities of three particles with linearly independent charges ''B'', ''L'' and ''Q''. In general one assigns the parity of the most common massive particles, the [[proton]], the [[neutron]] and the [[electron]], to be +1.
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| [[Steven Weinberg]] has shown that if '''P'''<sup>2</sup> = (−1)<sup>F</sup>, where '''F''' is the [[fermion]] [[number operator]], then, since the fermion number is the sum of the lepton number plus the baryon number, ''F''=''B''+''L'', for all particles in the Standard Model and since lepton number and baryon number are charges ''Q'' of continuous symmetries ''e''<sup>''iQ''</sup>, it is possible to redefine the parity operator so that '''P'''<sup>2</sup> = 1. However, if there exist [[Majorana fermion|Majorana]] [[neutrino]]s, which experimentalists today believe is quite possible, their fermion number is equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana, and so (−1)<sup>F</sup> would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity ±''i''.
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| ===Parity of the pion===
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| In 1954, a paper by William Chinowsky and [[Jack Steinberger]] demonstrated that the [[pion]] has negative parity.<ref>
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| {{cite journal
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| |last1=Chinowsky |first1=W.
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| |last2=Steinberger |first2=J.
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| |year=1954
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| |title=Absorption of Negative Pions in Deuterium: Parity of the Pion
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| |journal=[[Physical Review]]
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| |volume=95 |issue= 6|pages=1561–1564
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| |bibcode=1954PhRv...95.1561C
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| |doi=10.1103/PhysRev.95.1561
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| }}</ref> They studied the decay of an "atom" made from a [[deuteron]] ({{Nuclide2|hydrogen|2|charge=+|link=yes}}) and a negatively charged pion ({{subatomic particle|pion-}}) in a state with zero orbital [[angular momentum]] ''L'' = 0 into two [[neutron]]s (n).
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| Neutrons are [[fermion]]s and so obey [[Fermi–Dirac statistics]], which implies that the final state is antisymmetric. Using the fact that the deuteron has spin one and the pion spin zero together with the antisymmetry of the final state they concluded that the two neutrons must have orbital angular momentum ''L'' = 1. The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function (−1)<sup>''L''</sup>. Since the orbital momentum changes from zero to one in this process, if the process is to conserve the total parity then the products of the intrinsic parities of the initial and final particles must have opposite sign. A deuteron nucleus is made from a proton and a neutron, and so using the aforementioned convention that protons and neutrons have intrinsic parities equal to +1 they argued that the parity of the pion is equal to minus the product of the parities of the two neutrons divided by that of the proton and neutron in the deuteron, (−1)(1)<sup>2</sup>/(1)<sup>2</sup>, which is equal to minus one. Thus they concluded that the pion is a [[pseudoscalar particle]].
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| ===Parity violation===<!-- This section is linked from [[Tsung-Dao Lee]] -->
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| {{multiple image
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| | direction = vertical
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| | image1 = Parity clocks - P-conservation.svg<!-- file name only; no "File:" or "Image:" prefix -->
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| | width1 = 300
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| | image2 = Parity clocks - P-violation.svg<!-- file name only; no "File:" or "Image:" prefix -->
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| | width2 = 300
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| | footer = '''Top:''' P-symmetry: A clock built like its mirrored image will behave like the mirrored image of the original clock.<br>'''Bottom:''' P-asymmetry: A clock built like its mirrored image will ''not'' behave like the mirrored image of the original clock.
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| }}
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| Although parity is conserved in [[electromagnetism]], [[strong interactions]] and [[gravity]], it turns out to be violated in [[weak interactions]]. The Standard Model incorporates '''parity violation''' by expressing the weak interaction as a [[chirality (physics)|chiral]] gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in weak interactions in the [[Standard Model]]. This implies that parity is not a symmetry of our universe, unless a [[Mirror matter|hidden mirror sector]] exists in which parity is violated in the opposite way.
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| It was suggested several times and in different contexts that parity might not be conserved, but in the absence of compelling evidence these suggestions were not taken seriously. A careful review by theoretical physicists [[T. D. Lee|Tsung Dao Lee]] and [[C. N. Yang|Chen Ning Yang]]<ref>
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| {{cite journal
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| |last1=Lee |first1=T. D.
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| |last2=Yang |first2=C. N.
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| |year=1956
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| |title=Question of Parity Conservation in Weak Interactions
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| |journal=[[Physical Review]]
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| |volume=104 |issue=1 |pages=254–258
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| |bibcode = 1956PhRv..104..254L
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| |doi=10.1103/PhysRev.104.254
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| }}</ref> went further, showing that while parity conservation had been verified in decays by the [[strong force|strong]] or [[electromagnetic interaction]]s, it was untested in the [[weak interaction]]. They proposed several possible direct experimental tests. They were almost ignored, but Lee was able to convince his Columbia colleague [[Chien-Shiung Wu]] to try it. She needed special [[cryogenic]] facilities and expertise, so the [[The Wu Experiment|experiment]] was done at the [[National Bureau of Standards]].
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| In 1957 C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson found a clear violation of parity conservation in the beta decay of [[cobalt-60]].<ref>
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| {{cite journal
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| |last1=Wu |first1=C. S.
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| |last2=Ambler |first2=E
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| |last3=Hayward |first3=R. W.
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| |last4=Hoppes |first4=D. D.
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| |last5=Hudson |first5=R. P.
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| |year=1957
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| |title=Experimental Test of Parity Conservation in Beta Decay
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| |journal=[[Physical Review]]
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| |volume=105 |issue=4 |pages=1413–1415
| |
| |bibcode=1957PhRv..105.1413W
| |
| |doi=10.1103/PhysRev.105.1413
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| }}</ref> As the experiment was winding down, with double-checking in progress, Wu informed Lee and Yang of their positive results, and saying the results need further examination, she asked them not to publicize the results first. However, Lee revealed the results to his Columbia colleagues on 4 January 1957 at a "Friday Lunch" gathering of the Physics Department of Columbia.<!--ref>江才健 ''吳健雄: 物理科學的第一夫人'' p.216 時報文化出版企業股份有限公司 ISBN 957-13-2110-9</ref--> Three of them, [[Richard Garwin|R. L. Garwin]], [[Leon Lederman]], and R. Weinrich modified an existing cyclotron experiment, and they immediately verified the parity violation.<ref>
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| {{cite journal
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| |last1=Garwin |first1=R. L.
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| |last2=Lederman |first2=L. M.
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| |last3=Weinrich |first3=M.
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| |year=1957
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| |title=Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free Muon
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| |journal=[[Physical Review]]
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| |volume=105 |issue=4 |pages=1415–1417
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| |bibcode=1957PhRv..105.1415G
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| |doi=10.1103/PhysRev.105.1415
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| }}</ref> They delayed publication of their results until after Wu's group was ready, and the two papers appeared back to back in the same physics journal.
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| After the fact, it was noted that an obscure 1928 experiment had in effect reported parity violation in [[weak decay]]s, but since the appropriate concepts had not yet been developed, those results had no impact.<ref>
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| {{cite journal
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| |last1=Roy |first=A.
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| |year=2005
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| |title=Discovery of parity violation
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| |journal=[[Resonance (journal)|Resonance]]
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| |volume=10 |issue=12 |pages=164–175
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| |doi=10.1007/BF02835140
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| }}</ref> The discovery of parity violation immediately explained the outstanding [[Kaon#Parity violation|τ–θ puzzle]] in the physics of [[kaon]]s.
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| In 2010, it was reported that physicists working with the [[Relativistic Heavy Ion Collider]] (RHIC) had created a short-lived parity symmetry-breaking bubble in quark-gluon plasmas. An experiment conducted by several physicists including Yale's Jack Sandweiss as part of the STAR collaboration, suggested that parity may also be violated in the strong interaction.<ref>
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| {{cite web
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| |last=Muzzin |first=S. T.
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| |date=19 March 2010
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| |title=For One Tiny Instant, Physicists May Have Broken a Law of Nature
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| |url=http://www.physorg.com/news188211977.html
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| |work=[[PhysOrg]]
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| |accessdate=2011-08-05
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| }}</ref>
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| ===Intrinsic parity of hadrons===
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| To every particle one can assign an '''intrinsic parity''' as long as nature preserves parity. Although [[weak interaction]]s do not, one can still assign a parity to any [[hadron]] by examining the [[strong interaction]] reaction that produces it, or through decays not involving the [[weak interaction]], such as [[rho meson]] decay to [[pion]]s.
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| ==See also==
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| *[[Electroweak theory]]
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| *[[Standard Model]]
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| *[[Mirror matter]]
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| ==References==
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| ;General
| |
| *{{cite book
| |
| |title=Introduction to High Energy Physics
| |
| |last=Perkins
| |
| |first= Donald H.
| |
| |year= 2000
| |
| |ISBN= 9780521621960
| |
| }}
| |
| *{{cite book
| |
| |last1=Sozzi |first=M. S.
| |
| |year=2008
| |
| |title=Discrete symmetries and CP violation
| |
| |publisher=[[Oxford University Press]]
| |
| |isbn=978-0-19-929666-8
| |
| }}
| |
| *{{cite book
| |
| |last1=Bigi |first=I. I.
| |
| |last2=Sanda |first2=A. I.
| |
| |year=2000
| |
| |title=CP Violation
| |
| |series=Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology
| |
| |publisher=[[Cambridge University Press]]
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| |isbn=0-521-44349-0
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| }}
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| *{{cite book
| |
| |last1=Weinberg |first1=S.
| |
| |year=1995
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| |title=The Quantum Theory of Fields
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| |publisher=[[Cambridge University Press]]
| |
| |isbn=0-521-67053-5
| |
| }}
| |
| ;Specific
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| {{reflist}}
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| <!-- Categories -->
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| [[Category:Concepts in physics]]
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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| [[Category:Particle physics]]
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| [[Category:Nuclear physics]]
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| [[Category:Conservation laws]]
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